In this paper, we first develop the fractional Trudinger–Moser inequality in singular case and then we use it to study the existence and multiplicity of solutions for a class of perturbed fractional Kirchhoff type problems with singular exponential nonlinearity. Under some suitable assumptions, the existence of two nontrivial and nonnegative solutions is obtained by using the mountain pass theorem and Ekeland’s variational principle as the nonlinear term satisfies critical or subcritical exponential growth conditions. Moreover, the existence of ground state solutions for the aforementioned problems without perturbation and without the Ambrosetti–Rabinowitz condition is investigated.

在本文中，我们首先开发了奇异情况下的分数 Trudinger-Moser 不等式，然后我们用它来研究一类具有奇异指数非线性的扰动分数基尔霍夫型问题的解的存在性和多重性。在一些合适的假设下，利用山口定理和埃克兰变分原理作为非线性项满足临界或亚临界指数增长条件，得到两个非平凡非负解的存在性。此外，研究了在没有微扰和 Ambrosetti-Rabinowitz 条件的情况下上述问题的基态解的存在。